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Theorem elxp6 7709
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7613. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
elxp6 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Proof of Theorem elxp6
StepHypRef Expression
1 elxp4 7613 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
2 1stval 7677 . . . . 5 (1st𝐴) = dom {𝐴}
3 2ndval 7678 . . . . 5 (2nd𝐴) = ran {𝐴}
42, 3opeq12i 4773 . . . 4 ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩
54eqeq2i 2814 . . 3 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩)
62eleq1i 2883 . . . 4 ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵)
73eleq1i 2883 . . . 4 ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶)
86, 7anbi12i 629 . . 3 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))
95, 8anbi12i 629 . 2 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
101, 9bitr4i 281 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  {csn 4528  cop 4534   cuni 4803   × cxp 5521  dom cdm 5523  ran crn 5524  cfv 6328  1st c1st 7673  2nd c2nd 7674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-1st 7675  df-2nd 7676
This theorem is referenced by:  elxp7  7710  eqopi  7711  1st2nd2  7714  eldju2ndl  9341  eldju2ndr  9342  r0weon  9427  qredeu  15996  qnumdencl  16073  setsstruct2  16517  tx1cn  22218  tx2cn  22219  txhaus  22256  psmetxrge0  22924  xppreima  30412  ofpreima2  30433  smatrcl  31153  1stmbfm  31632  2ndmbfm  31633  oddpwdcv  31727  prproropf1olem0  44012
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